–1
1
2
y
–2
–1
1
2
x
–2
–1
0
1
2
y
–2
–1
1
2
x
Figure 3.9: The velocity fieldv2,for (a)z= 1and (b)z=−1, and the parametersl= 1,L= 1,e10=−1, k= 0.5,b0= 1, andκ= 1.
non-ideal region. There the ratio (ve/vAe) of the plasma velocity to the Alfv´en velocity is given by ve vAe = √ π √ 2Rme Le l L l g, (3.34)
whereRme is the global magnetic Reynolds number,Rme = LevAe/η, andg = exp (kLe−1/2)is a factor relating to the geometry of the magnetic field. The ordering of parametersLE > L > l has been
assumed. The parameterslandL, which relate to the structure of the non-ideal region, andg, which relates to the field geometry, would not in a general 3D reconnection event be arbitrary, but rather determined by the evolution of the magnetic field before the onset of a stationary phase. Therefore the expression (3.34) should be interpreted with particular care. Although at first sight it appears to scale asR−1
me, each of the other factors on the right hand side of (3.34) can be much larger than unity and also depend onRme (l=l(Rme),L=L(Rme),g =g(Le, Rme)). Determining howve/vAe scales withRme and so whether or not the reconnection is fast is outside the scope of this simple stationary model. Instead we proceed to examine the case of ‘composite solutions’.
3.3
Composite Solutions
In many realistic situations the plasma velocity outside the HFT will be non-zero, and therefore we here choose to superimpose an ideal solution (v16= 0) on the particular solution, giving composite solutions. In the kinematic analysis, as given by Hornig and Priest (2003), the two solutions are essentially independent, but in the present dynamic analysis they are coupled in the momentum equation (3.8) by the inertial term
(v1· ∇′)v1. We now examine the extent to which the coupling restricts the choice of the ideal solution, and investigate how the reconnection process differs between the particular and composite solutions.
3.3 Composite Solutions 40
In general, the momentum equation given by (3.8) implies a coupling between the ideal and non-ideal Ohm’s laws given by equations (3.4) and (3.5). However, for the class of ideal plasma flowsv1for which the curl of the inertial term on the left-hand side of (3.8) vanishes, the equations become decoupled. In this case the effects of a non-trivial solution to (3.4) are apparent at second order only in the pressure gradient ∇p2. For ideal flows satisfying this condition the particular solutions of Section 3.2 may be taken as a solution to (3.5), and so we have a direct comparison with the composite solutions of Hornig and Priest (2003). We begin by examining an ideal stagnation flowv1for which∇ ×(v1· ∇)v1 = 0. Stagnation flows are an obvious choice to consider because they can lead to the build-up of thin current sheets. They also allow for flux to be brought into and removed from the localised non-ideal region and so change field-line connectivities away from this region.
Turning first to (3.4), we takeφ0to be the function of the field line coordinates(x0, y0)given by
φ0= ϕ0
Λ2x0y0. (3.35)
SettingZ0= 0, the inverse field line mappings (3.13) can be used to find an equivalent expression in terms ofx,y, andz, so determiningφ0(x, y, z). The component ofv1perpendicular toB0may then be deduced from (3.4) as
v1⊥=−∇φ0×
B0 |B0|2
. (3.36)
We use the freedom in choosing the component ofv1parallel toB0to set thez-component ofv1to zero:
v1=v1⊥−
(v1⊥)zB0
b0 . (3.37)
This ensures the flow is divergence-free since the z-component of the curl of equation (3.4) becomes b0(∇ ·v1) = 0. Thus we obtain
v1= ϕ0
b0Λ2(xcosh(2kz)−ysinh(2kz))xˆ+ ϕ0
b0Λ2(xsinh(2kz)−ycosh(2kz))yˆ,
and see that∇ ×(v1· ∇)v1 = 0. The ideal flow crosses the separatrices ofB0 in thexy-plane, with streamlines ofv1above and below the central plane shown in Figure 3.10.
Since the inertial term in (3.8) may be expressed as the gradient of a scalar function, the equation has the same structure as in the case of the particular solutions (wherev1 = 0), which were examined in Section 3.2. Thus the same form ofj2may be taken in both the particular and composite case, and continue to assume the form given by
j2=
j20
cosh2x2λ−2y2 zˆ,
as in the Subsection 3.2.2. A further comparison with the numerical simulation of Pontin et al. (2005a) can now be made; our ideal stagnation flowv1in the central plane has a similar profile to the plasma flow at the end of their simulation when viewed with the correct orientation according to the current concentration.
3.3 Composite Solutions 41
−4
−2
2
4
−4
−2
2
4
y
x
(a)−4
−2
2
4
−4
−2
2
4
y
x
(b)Figure 3.10: The ideal plasma velocityv1for (a)z = 0.5and (b)z =−0.5, and the parametersϕ0 = 1, k= 0.5,b0= 2, andΛ = 1.
Using (3.8) we deduce the pressure termp2as
p2=p20−1 2kλ 2b0j20tanh x2−y2 λ2 − ϕ 2 0 2Λ2b2 0 x2+y2 Λ2 . (3.38)
The particular solution may be recovered by settingϕ0= 0, and so it is seen that the inclusion of a zeroth- order flow has had the effect of introducing an extra term to the pressure, proportional toϕ2
0/(Λ4b20). When ϕ0 = 0there are strong gradients in the pressure along the separatrices ofB0in thexy-plane. This extra term has the effect of smoothing out these strong gradients, withp2 becoming a smoother function as ϕ2
0/Λ2b20is increased. An example of the pressure profile is shown in Figure 3.11, which can be compared with Figure 3.5 of Section 3.2. The additional term has a natural physical explanation. It deflects the incomingv1flow toward the outflow direction, a purely hydrodynamical effect. Due to the symmetry with respect to inflow and outflow, there is no net transfer of magnetic energy to kinetic (bulk) energy of the plasma in this stationary solution, as would be expected in a more realistic situation. However, we may model part of this process by requiringv1·j2×B0to be positive. This would result in an initial transfer of magnetic energy to kinetic energy, but with the latter subsequently transferred to potential energy, since
v· ∇p >0, so no net acceleration can take place. We have here that
v1·j2×B0=−
ϕ0j20k x2+y2cosh (2kz)−2xysinh (2kz)
Λ2cosh2x2−y2
λ2
.
We require this quantity to be positive, which can be ensured by taking the combinationϕ0j20k <0.
Now turning to (3.5) we may use the same quantitiesη,ˆ E1, andv2as in Section 3.2.2 to satisfy the equation.
3.3 Composite Solutions 42 –4 –2 0 2 4 x –5 0 5 y 0.5 1 1.5 2
Figure 3.11: The pressure profilep2(x, y) whenv1 6= 0,j2 = j20/cosh2 x2−y2
/λ2zˆ,and the parametersp20 = 2,b0= 2,λ= 1,k= 0.5,j20 =−1,Λ = 1andϕ0 = 2. The lower pressure regions correspond to inflow ofv1and the higher pressure regions to outflow.
The question that arises at this point then is: how do the particular and composite solutions differ? Or, in other words, what physical effect does the inclusion of the ideal flowv1have on the solution? Since
E0 is perpendicular toB0the expression for the rate of reconnection is the same as that of the particular solution. However the non-vanishing external flow changes the meaning of this reconnection rate since the reconnection process can now reconnect flux outside the hyperbolic flux tube. The evolution of magnetic flux in the two cases is therefore quite different, and may be visualised using the concept of a magnetic flux velocity as described in Section 2.1. We demonstrate in the following subsection how the magnetic flux evolution differs between the particular and the composite solutions.
Magnetic flux that does not pass through the diffusion region evolves ideally, i.e. it is frozen into the flow and so initially-connected plasma elements remain connected. We may track the evolution of plasma elements in the ideal flow above and below the diffusion region. Initially-connected elements will only remain connected if the field line linking them does not pass through the non-ideal region; otherwise the elements will change their connectivity. The pair of quasi-flux velocitieswinandwout can be used to
project into the central plane (z = 0) the flow lines corresponding to the ideal evolution above and below the non-ideal region. Examining the differences between the lines ofwinandwoutallows us to deduce
how the magnetic flux evolves.
For the stagnation flow described in Section 3.3(a), the relevant projection is shown in Figure 3.12 (for a particular choice of parameter values). The flow lines ofwin(grey lines) in thez = 0plane are
superimposed on those ofwout (black lines) in the same plane. We are able to divide the plane into three
regions according to the type of reconnective behaviour that occurs; the separatrices dividing these regions are shown in Figure 3.13.
3.3 Composite Solutions 43 –1.5 –1 –0.5 0 0.5 1 1.5 y –1.5 –1 –0.5 0 0.5 1 1.5 x
Figure 3.12: Flowwin(grey) andwout(black) for the solution already described in Section 3.3.
–1.5 –1 –0.5 0 I II III 0.5 1 1.5 y –1.5 –1 –0.5 0 0.5 1 1.5 x
Figure 3.13: Separatrices ofwin(grey) andwoutblack for the solution described in Section 3.3. The region
is divided into three types of reconnective behaviour. Magnetic flux passing through region I undergoes ideal evolution. Magnetic flux passing through region II undergoes a slippage-like behaviour while flux passing through region II undergoes an evolution similar to that seen in classical 2D reconnection.
3.3 Composite Solutions 44
z= 0plane in region I evolves ideally, so that initially-connected plasma elements will remain connected. In regions II and III the flow lines ofwinandwoutdo not coincide. For magnetic flux passing through the
z = 0plane in these two regions we deduce that plasma elements above and below the non-ideal region that are initially connected will not remain so. Tracking the evolution of corresponding pairs allows us to distinguish different types of magnetic flux evolution.
Magnetic flux passing through region II exhibits a slippage-like behaviour. Initially connected plasma elements above and below the non-ideal region will change their connections as the flow transports the magnetic flux linking them into the non-ideal region. On leaving the shadow of the non-ideal region the ini- tially connected elements are both transported in the same direction by the flow and a new ideal connection is again established for each plasma element. Although this connnection will not be with the initial partner, it will be with a plasma element that was initially close to that partner.
Magnetic flux passing through region III exhibits the type of behaviour most similar to that shown in classical 2D reconnection. Again, initially-connected plasma elements above and below the non-ideal region loose their connections as the magnetic flux linking them is transported into the non-ideal region. However, on leaving the shadow of the non-ideal region the initially-connected plasma elements above and below the non-ideal region are transported in different directions by the flow, along opposing ‘wings’ seen in Figure 3.13, and their separation will therefore increase in time, as in the classical 2D reconnection picture. The new ideal connection for a plasma element initially above (below) the non-ideal region will be with a plasma element that was initially below (above) the non-ideal region in the distant opposing wing.
Therefore in this composite solution the stagnation flow is dominant, with the rotational flowv2present as a background flow. A stagnation flow was found to develop in the numerical simulations of Pontin et al. (2005a), although a background counter-rotational rotational flow was also shown to be present, and seen to fall off with distance from the X-line. The simulation also confirmed a continual and continuous change of field line connectivity. Thus many properties of their simulation are captured in the above-described analytical solution.
We have been able to make a direct comparison between the particular solutions described in Section 3.2 and the composite solutions described in this section since, for our choice ofv1, the curl of the inertia term in (3.8) vanishes. In principle we could have chosen other ideal flows for this directly comparable analysis that also have a curl-free inertial term. One example is that which results from defining the scalar function φ0as the function of field-line coordinates given by
φ0=ϕ0 Λ2 x 2 0−y02 , from which we obtain
v1= −
2ϕ0
b0Λ2 (yxˆ+xzˆ). (3.39) This is also a stagnation flow, but it differs considerably from the flow considered in the previous section; it does not cross the separatrices of the projection ofB0onto thexy-plane, and is independent of the third coordinate,z. When superimposed on the particular solution, however, the same three regions of differing flux evolution are present, as illustrated in Figure 3.14. The inflow and outflow channels bounded by the separatrices of the quasi-flux velocities are now centred around the separatrices ofB0in thez= 0plane.
3.3 Composite Solutions 45 –2 –1 0 1 2 y I II III –2 –1 0 1 2 x
Figure 3.14: Separatrices ofwin(grey) andwout black whenv1is given by (3.39). The same three types of reconnective behaviour as figure 3.13 are present.
This demonstrates one of the crucial differences between 2D and 2.5D reconnection and the 3D case. The crossing of the separatrices by the flow is only a criterion for reconnection in the 2D case. In 3D the difference betweenwinandwoutis the crucial property for reconnection. Another example in this class of
flows which can be used to form composite solutions is the rotational ideal flow arising from the choice
φ0=ϕ0 Λ2 x 2 0+y02 . (3.40)
This ideal flow is rotating in the same sense for allz, and so does not have the effect of bringing flux into and away from the non-ideal region.
For the three flows examined in this section, the reconnection rate, as determined by the integral of the parallel electric field along the reconnection line, is identical, but the magnetic flux evolution quite different. The distinct types of reconnective behaviour illustrated here, and in paper I, may be distinguished by considering the associated internal and external reconnection rates, as introduced by Hornig (2006).
The external reconnection rate measures the rate at which flux is transported into (and equivalently out of) the non-ideal region. This rate is always less than or equal to the total reconnection rate, and the internal
reconnection rate measures the difference between the total reconnection rate and the external reconnection
rate. The electric potential along the flow lines ofwin andwout is constant, since these are streamlines
of the ideal flow. The difference in electric potential between the inflow channels bounded by the separa- trices of the flow therefore quantifies the external reconnection rate, while the internal reconnection rate is given by the difference between the total rate (measured by the integrated parallel electric field along the reconnection line) and the external reconnection rate.